A Bitcoin wallet is what you use to receive, store and transfer your bitcoins.
Setting up a Bitcoin wallet is very easy and takes just a few seconds. All you need is an email address and a password of your choice. You will be able to send and receive payments immediately.
You can think of your wallet as your bank account. Just like a bank account which has an assigned, unique bank account number, your Bitcoin wallet has an assigned, unique wallet address. This address consists of up to 34 random digits (consisting of numbers and letters, both uppercase and lowercase, usually starts with 1 or 3). Here's an example of a Bitcoin wallet address:
There are four different kinds of wallets: desktop wallets, browser-based wallets, mobile wallets, and hardware wallets. They differ mainly in where they are stored (i.e. either on your laptop, on a third party website, on a mobile device, or on a specialized offline device), and thus, they each come with their own pros and cons.
Click here for a highly informative article on the four different kinds of wallets.
Click here for a list of available Bitcoin wallet services.